Contact Geometry, Fano Orbifolds, and Einstein Metrics

接触几何、Fano Orbifolds 和爱因斯坦度量

• 批准号: 0203219
• 负责人: Charles Boyer
• 金额: $ 31.5万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-15 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203219&HistoricalAwards=false
• 关键词: Contact; Geometry; Fano; Orbifolds; Einstein

ABSTRACT DMS - 0203219.Professors Boyer and Galicki propose to investigate several projects in geometry and topology. The objective of all the projects is to study fundamental questions in Riemannian Geometry with two main focal points: Contact Geometry oforbifold bundles over Fano varieties and the existence of somespecial (i.e., Einstein, positive Ricci curvature) metrics on such spaces. The questions and problems proposed here are deeply rooted in theprincipal investigators' earlier work which exploited a fundamental relationship between contact geometry of Sasakian-Einstein spaces and two kinds of Kaehler geometry, namely Q-factorial Fano varieties with Kaehler-Einstein orbifold metrics with positive scalar curvature, and Calabi-Yau manifolds with their Kaehler Ricci-flat metrics. The most recent work of the principal investigators has led to an important breakthrough in the study of such structures when,using recent results of Demailly and Kollar, the principal investigatorswere able to construct new examples of compact Einstein manifolds in dimension 5 as well as many positive Einstein metrics on families of rationalhomology 7-spheres. The techniques used by the principal investigators borrow fromseveral different fields; the algebraic geometry of Mori theory and intersectiontheory, the analysis of the Calabi Conjecture, and finally theclassical differential topology of links of isolated hypersurfacesingularities. These methods can be extended much further and invarious directions. An example that stands out in this respect is the question of the existence of Einstein metrics on exotic spheres, which is one of the main objectives of this proposal. More generally the principal investigators want to address several classification problems concerning compact Sasakian-Einstein manifolds in dimensions 5 and 7. These two dimensions are important for two separate reasons. In view of earlier work higher dimensional examples can be constructed using the join construction. At the same time these two odd dimensions appear to play special role in Superstring Theory. In the context of recent developements inString and M-Theory the principal investigators also propose toinvestigate some related problems concerning self-dual Einstein metrics in dimension 4 and exceptional holonomy metrics in dimension 7 and 8.This project is intended to further the understanding of the mathematicsof certain important types of geometry. While this endeavor is notdirectly motivated by advances in technology, the history of mathematicsadequately demonstrates that today's pure mathematics often becomestomorrow's applied mathematics. Indeed the mathematics being considered inthis project is currently being used in the field of Elementary ParticlePhysics, more specifically, in the attempts at understanding a unifieddescription of the fundamental forces of the universe.-

摘要DMS - 0203219。Boyer教授和Galicki教授提议研究几何学和拓扑学方面的几个项目。所有项目的目标都是研究黎曼几何中的基本问题，主要有两个焦点：Fano变体上的双折束的接触几何和这些空间上某些特殊度量（即爱因斯坦，正Ricci曲率）的存在性。这里提出的问题和问题深深植根于主要研究者早期的工作，该工作利用了sasaki - einstein空间的接触几何与两种Kaehler几何之间的基本关系，即具有正标量曲率的Kaehler- einstein轨道度量的q ! Fano变体和具有Kaehler- ricci -平坦度量的Calabi-Yau流形。主要研究人员最近的工作在这种结构的研究中取得了重大突破，利用Demailly和Kollar的最新成果，主要研究人员能够在5维构造紧致爱因斯坦流形的新例子，以及在理性同调7球族上的许多正爱因斯坦度量。主要研究人员使用的技术借鉴了几个不同的领域；森理论和交点理论的代数几何，卡拉比猜想的分析，最后是孤立超曲面奇点链路的经典微分拓扑。这些方法可以进一步向各个方向推广。在这方面，一个突出的例子是爱因斯坦度规在奇异球体上的存在性问题，这是本提议的主要目标之一。更一般地说，主要研究人员想要解决关于5维和7维的紧化sasaki - einstein流形的几个分类问题。这两个维度之所以重要，有两个不同的原因。鉴于先前的工作，高维的例子可以使用连接构造来构造。同时，这两个奇维似乎在超弦理论中扮演着特殊的角色。在弦理论和m理论的最新发展背景下，主要研究人员还提出了有关4维自对偶爱因斯坦度量和7维和8维异常完整度量的一些相关问题。这个项目的目的是进一步理解某些重要几何类型的数学。虽然这一努力并非直接受到技术进步的推动，但数学的历史充分证明，今天的纯数学往往会成为明天的应用数学。事实上，在这个项目中考虑的数学目前正在基本粒子物理学领域中使用，更具体地说，是在试图理解对宇宙基本力的统一描述